Remark 2.5 This iterative bridge deletion procedure admits a variant which does not
look at \(M\): for \(G = (V,E)\) , choose a bridge \((u,v) \in E\)  such that the
connected components of \(u\)  and \(v\)  in \((V, E \setminus \lbrace (u,v)\rbrace )\)  have an
odd number of vertices (equivalently, the two new connected components in \(G[V\setminus \lbrace u,v\rbrace ]\)  disconnected by the removal of \(u\)  and \(v\)  have an even
number of vertices). If, by doing so iteratively, one reaches the empty graph,
then the set of deleted bridges is the unique perfect matching of \(M\) . This
gives a quasi-linear time algorithm for finding a unique perfect
matching, see [1]} and
[2]}.

